[FOM] Inconsistency of P

Panu Raatikainen panu.raatikainen at helsinki.fi
Wed Oct 5 02:46:42 EDT 2011

"Daniel Mehkeri" <dmehkeri at gmail.com>:

>> (though, as I emphazides, S  is then not a subtheory of T)
> I still don't understand why.

I think we have now sorted this out with Monroe Eskew in a couple of  
off-list e-mails.

Focusing on the content of Chaitin's theorem rather than on any  
particular proof of it, I assumed that c is a finite limit such that T  
cannot prove K(n) > c.

But that, of course, tacitly presupposes that T is really consistent.

So I thought the inconsistency of T was only hypothetical: "If T  
proved K(n)>c, then it would be inconsistent".

I did not take the alternative that T *really* is inconsistent  
seriously, because then c, understood as above, would not even be  

* * *

If c, on the other hand, is only a suitable constant which is involved  
in a Chaitin machine construction for T (inconsistent), then a  
consistent subsystem S of T may prove K(n) > c.

Or, at least, it is not clear to me it could not.

This case just is not very interesting, because *all* consistent  
theories (in the language of T) are subsystems of T; and trivially,  
some of these prove  K(n) > c (let c be however large and complex).



Panu Raatikainen

Ph.D., University Lecturer
Docent in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki

E-mail: panu.raatikainen at helsinki.fi


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